GCM, GHASH and Weak Keys

The Galois/Counter Mode (GCM) of operation has been standardized by NIST to provide single-pass authenticated encryption. The GHASH authentication component of GCM belongs to a class of Wegman-Carter polynomial universal hashes that operate in the field GF (2). GCM uses the same block cipher key K to both encrypt data and to derive the generator H of the authentication polynomial. In present literature, only the trivial weak key H = 0 has been considered. In this note we show that GHASH has much wider classes of weak keys in its 512 multiplicative subgroups, analyze some of their properties, and give experimental results when GCM is used with the AES algorithm.